Just the same as spewing water from the back of jets on a clear day can cause changes in the atmosphere and hence, modify the weather, by producing water vapour clouds.
NO THEY DON'T. The largest change in insolation ever measured (measuring a sky overcast by trails) involved less than a 2% change. But you lose the blue, of course.

Just look at the sky on a clear day when the jets are out spraying in criss-cross patterns.
Routes cross each other, and you are a simpleton. The stratosphere travels in a different direction from your prevailing wind, has an average speed of 50+mph. Crossing shuttle routes will produce a "grid" if the stratospheric layers they pass through approach saturation with water vapor. You cannot SEE the stratosphere, wet or dry. Grids can be made over your horizon and yet arrive overhead an hour or two later.

Soon, the long white streaks fan out into thin, but wide, white clouds.
All made out of 99.9994% pure water...

Take note as to what type of weather follows.
Watch my T-shirt: the persistent spreading contrail starts by being entrained in the aircraft's wing vortices. These are angled shallowly downwards such that by the time they have stopped spinning the trail is 200 miles behind and a thousand feet beneath the aircraft. Each vortex loop forms a serration in the trail up to two thousand feet deep. Each serration is actually cone-shaped, with a centre of larger ice particles falling fastest. The faster they fall, the faster they accrete ice, so it is a form of instability.
Normally before this happens the saturated layer has ceased to be the ambient layer, and the ice starts to evaporate again. If it makes it down through the tropopause then it will evaporate as it falls through the ever-warming air. This gives you the high leaden overcast.
If layers have a LATERAL motion then a contrail will SHEAR in a cirrus-like manner - because, technically it IS a cirrus cloud, its only distinction being that of possessing a slightly-different granularity - of ICE.